**4.1. Electrical tomography**

The solution to the electrical impedance tomography (EIT) problem is to determine the potential distribution in the region Ω in given boundary conditions and full information about the analyzed region [19–21]. In order to obtain benefits, the accuracy of EIT has historically been divided into capacity (ECT), for systems dominated by dielectrics and resistance (ERT), for conducting processes. The basic theory can be obtained from Maxwell's equations. A complex "admittivity" is defined as follows:

$$
\gamma = \sigma + \text{ioos} \tag{1}
$$

where σ is the electrical conductivity, ε is the permittivity, and ω is the angular frequency. For an electric field strength (Ε), the free current density (J) in the area under investigation will be related by Ohm's law:

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$$\mathbf{J} = \mathbf{y} \mathbf{E} \tag{2}$$

The gradient of the potential distribution (u) is as follows:

$$E = -\nabla u \tag{3}$$

Assuming there are no current sources within the examined region then from Ampère's law:

$$
\nabla \cdot \mathbf{J} = \mathbf{0} \tag{4}
$$

The potential distribution in the isotropic, inhomogeneous region is as follows:

$$\nabla \cdot (\mathcal{V} \cdot \nabla u) = 0,\tag{5}$$

where *u* is the potential.

studied environment. Such measurements do not change physical and chemical parameters. In the reconstruction of the image, the key parameters are the speed of analysis of flowing raw materials and the accuracy of reconstructed processes. The measurement must be fast because some industrial processes run at high speed. The measuring system consists of a sensor, specialized electronics for capacitance measurement, and data reconstruction and analysis system. Industrial tomography applications are usually a challenge for obtaining spatial distribution data from observations that go beyond the process boundary. The biggest challenge is to achieve effective coverage of closed spaces using practical resources at a reasonable cost. Sensor networks with feedback loops are the basic elements of production control. Distributed infrastructure requires various tasks related to detection and startup and is usually characterized by internal spatial organization. The decisive difference in the mass production of chemicals, food, and other goods is that joint process sensors only provide local measurements. In most production systems, such local measurements are not representative of the entire process; therefore, spatial solutions are necessary. The general measurement

The solution to the electrical impedance tomography (EIT) problem is to determine the potential distribution in the region Ω in given boundary conditions and full information about the analyzed region [19–21]. In order to obtain benefits, the accuracy of EIT has historically been divided into capacity (ECT), for systems dominated by dielectrics and resistance (ERT), for conducting processes. The basic theory can be obtained from Maxwell's equations. A complex

*γ* = *σ* + i (1)

where σ is the electrical conductivity, ε is the permittivity, and ω is the angular frequency. For an electric field strength (Ε), the free current density (J) in the area under investigation will be

model for tomographic sensors is shown in **Figure 4**.

**Figure 4.** General measurement model for tomography sensors.

30 New Trends in Industrial Automation

**4. Methods and models**

**4.1. Electrical tomography**

related by Ohm's law:

"admittivity" is defined as follows:

As above the ratio of ωε/σ, when the capacitance or the resistance term is dominant, the governing equation is further simplified:

$$\nabla \cdot (\sigma \,\,\nabla u) = 0 \, f \sigma \, \frac{a\epsilon}{\sigma} \, \ll 1 \, (\text{ERT}) \tag{6}$$

$$\nabla \cdot (\varepsilon \,\,\,\nabla u) = 0 \,\, for \,\frac{\omega \varepsilon}{\sigma} \gg 1 \,\, (\text{ECT})\tag{7}$$

As a result of the inverse problem solution, the conductivity distribution in the tested area is obtained.

A set of electric currents are injected into the examined object through these electrodes, and the obtained voltages are measured using the same electrodes. **Figure 5** shows the opposite method of acquiring boundary potential data illustrated for a cylindrical volumetric guide and 16 equally spaced electrodes: (a) first measurement and (b) second measurement.

In electrical capacitive tomography, the information source is the electrical capacitance between the electrodes located on the perimeter of the measurement sensor (see **Figure 6**). An important feature of the measurement is the non-invasive contact of the sensor with the tested object. Such a solution does not interfere with industrial processes. The advantage of this technique is the quick acquisition of measurement data. The laboratory measuring system with sensors is shown in **Figure 7**.

#### **4.2. Ultrasound tomography**

Measurement methods using information contained in the ultrasonic signal after passing through the medium under investigation are called ultrasound transmission methods (see **Figure 8**). Ultrasonic waves belong to short waves and have propagating and radiation properties. The length of these waves depends on the medium to which they are emitted

**Figure 5.** Opposite method in electrical resistance tomography.

and is in the range from a few micrometers in liquids to several dozen centimeters in metals. They can be used to measure the attenuation coefficient and ultrasonic time of signal transitions in the medium subjected to their influence. Many measurements can be made using ultrasound without fear of damage or irradiation of the examined objects. This technique allows to obtain quantitative images of the internal structure, in which the numerical values of each pixel describe such physical properties of the examined objects, e.g., temperature distribution, density, or viscosity. Measurements of parameters such as signal transitions,

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**Figure 7.** Laboratory measurement system.

**Figure 8.** Idea of measurement model in ultrasound tomography.

**Figure 6.** Measurement model of electrical capacitance tomography.

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**Figure 7.** Laboratory measurement system.

**Figure 8.** Idea of measurement model in ultrasound tomography.

**Figure 6.** Measurement model of electrical capacitance tomography.

**Figure 5.** Opposite method in electrical resistance tomography.

32 New Trends in Industrial Automation

and is in the range from a few micrometers in liquids to several dozen centimeters in metals. They can be used to measure the attenuation coefficient and ultrasonic time of signal transitions in the medium subjected to their influence. Many measurements can be made using ultrasound without fear of damage or irradiation of the examined objects. This technique allows to obtain quantitative images of the internal structure, in which the numerical values of each pixel describe such physical properties of the examined objects, e.g., temperature distribution, density, or viscosity. Measurements of parameters such as signal transitions, attenuation coefficient, and its derivative through frequency allow after appropriate reconstructive transformations, imaging of the internal structure of the tested medium, as well as flow parameters such as its speed, average speed, or profile speed. The basis for imaging is differences in local values of specific acoustic parameters. The image obtained by means of appropriate reconstruction methods presents the distribution (obtained from the measurement data using the scanning technique from as many directions as possible after the ultrasonic pulses have passed through the tested environment) [22].

The problem of image construction in the case of ultrasound very often leads to the overdetermined algebraic set of equations that can be expressed in the matrix form:

$$\mathbf{W}f = \mathbf{s},\tag{8}$$

finite elements. The numerical experiment was carried out on noisy data. **Figure 11** presents

**Figure 10.** Example of the image reconstruction (ECT). (a) Levenberg-Marquardt method and (b) modified Levenberg-

**Figure 9.** Image reconstruction (ERT): (a) Gauss-Newton with Laplace regularization and (b) Gauss-Newton with

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**Figure 12** shows images of the experiments by ultrasound tomography. The algorithm was designed in such a way that an overdetermined system of equations could be generated, i.e., one for which the number of equations is greater than the number of unknowns. A feature of the tomography is, among other things, that the coefficient matrix is a rectangular deficiency of the pseudo-rank matrix. In such cases, you should consider trial solutions and choose only one of them. The obtained results are a raw tomographic image for synthetic data. In the numerical experiments presented, no additional adjustment method was used to obtain

the image reconstruction by elastic net method in ECT.

Tikhonov regularization (ERT).

Marquardt method.

where *W* is the matrix of dimensions m x n and m > n, *s* is the right-hand side vector (one column matrix), and *f* is the solution vector.

One of the ways of the solution (Eq. (7)) is to find the vector *f* \* , which minimizes Euclidean norm of residual vector *r* for the known matrix *W* and vector s, and it means:

$$\|r\|\_{\ast} = \min \|\mathbf{s} - \mathbf{W}f\|\_{\ast}, \quad \|f^{\ast}\|\_{\ast} = \min \|f\|\_{\ast} \tag{9}$$

where the last minimum is taken for all vectors *f* which fulfill the previous relation. Equation (8) is well known as a linear least squares problem (LSP).
